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Let $𝐾$ be a field and $𝐿,{𝐿}^{\prime }$ two finite extensions of $𝐾$.

(i) Prove that if $𝐿/𝐾$ is separable then $𝐿{\otimes }_{𝐾}{𝐿}^{\prime }$ is isomorphic to a product ${\prod }_{𝑖=1}^{𝑟}{𝐿}_{𝑖}^{\prime }$ of finitely many fields. (Hint: use the primitive element theorem to write $𝐿=𝐾[𝑡]/(𝑓)$ for a suitable monic $𝑓$.)

(ii) Give an example to show this need not be true without the separability assumption.

Suppose $𝑃$ is a prime ideal of $ℂ[𝑥,𝑦]$. Show that $𝑃=(0)$, or $𝑃=(𝑓)$ where $𝑓$ is an irreducible polynomial, or $𝑃=(𝑥-𝑎,𝑦-𝑏)$, where $𝑎$ and $𝑏$ are complex numbers.

Hint. The following intermediate step may be useful. If $𝑃$ contains two irreducibles $𝑓,𝑔$, not multiples of each other, use the Euclidean algorithm in $ℂ(𝑦)[𝑥]$ to find a nonzero $ℎ\in (𝑓,𝑔)$ so that $ℎ\in ℂ[𝑦]$.

Let $𝐴$ be a commutative ring. We say an $𝐴$-module $𝑀$ is finitely presented if it is isomorphic to the cokernel of an $𝐴$-module homomorphism ${𝐴}^{\oplus 𝑚}\to {𝐴}^{\oplus 𝑛}$ for some $𝑚,𝑛\ge 1$.

(i) Let $𝐵$ be a flat $𝐴$-algebra, $𝑀$ a finitely presented $𝐴$-module, and $𝑁$ any $𝐴$-module. Prove the natural map

is an isomorphism.

(ii) Suppose

is a short exact sequence of $𝐴$-modules. Suppose that for each maximal ideal $𝔪\subset 𝐴$, the localized sequence

is split. If ${𝑀}^{″}$ is finitely presented, show that $(\ast )$ is split.

Hint: Prove $(\ast )$ splits if and only if the map ${Hom}_{𝐴}({𝑀}^{″},𝑀)\to {Hom}_{𝐴}({𝑀}^{″},{𝑀}^{″})$ induced by the right map $𝑀\to {𝑀}^{″}$ of $(\ast )$ is surjective.

Let $𝑓={𝑋}^3-2\in \mathbb{Z}[𝑋]$.

(i) Prove $𝑓$ is irreducible over $\mathbb{Q}$ and that its splitting field $𝐾/\mathbb{Q}$ has Galois group ${𝑆}_3$.

(ii) For each subgroup $𝐻$ of ${𝑆}_3$, determine with proof the corresponding field ${𝐾}^{𝐻}$.

(iii) Prove that the splitting field of $𝑓$ over ${𝔽}_5$ is quadratic and over ${𝔽}_7$ is cubic.

Let $𝐹$ be a field, and let $𝑉$ be the 4-dimensional vector space ${𝐹}^4$ with the skew-symmetric bilinear form

A subspace $𝑈$ of $𝑉$ is isotropic if $⟨{𝒖}_1,{𝒖}_2⟩=0$ for ${𝒖}_1,{𝒖}_2\in 𝑈$. For example, the two-dimensional subspace ${𝑈}_0$ of $𝒙=({𝑥}_1,{𝑥}_2,{𝑥}_3,{𝑥}_4)$ with ${𝑥}_3={𝑥}_4=0$ is isotropic. Let $𝐺$ be the group of $𝑔\in {GL}_4(𝐹)$ such that

for all $𝒙,𝒚\in 𝑉$.

(a) Prove that if $𝑈$ is a two-dimensional isotropic subspace then there exists $𝑔\in 𝐺$ such that $𝑔𝑈={𝑈}_0$. Hint: let ${𝒖}_1,{𝒖}_2$ be any basis of $𝑈$. Show that there exist ${𝒗}_1$ and ${𝒗}_2$ such that $⟨{𝒗}_1,{𝒗}_2⟩=0$ and $⟨{𝒖}_{𝑖},{𝒗}_{𝑗}⟩={𝛿}_{𝑖𝑗}$.

(b) Suppose that $𝐹={𝔽}_{𝑞}$. Show that the number of two dimensional isotropic subspaces of $𝑉$ is $({𝑞}^2+1)(𝑞+1)$. Hint: First count the number of pairs $({𝒖}_1,{𝒖}_2)$ so that ${𝒖}_1,{𝒖}_2$ are linearly independent and $⟨{𝒖}_1,{𝒖}_2⟩=0$.

Let $𝐺=GL(2,{𝔽}_{𝑞})$, where $𝑞={𝑝}^{𝑛}$, $𝑝$ prime. Let $\mathrm{\Pi }$ be the set of one-dimensional subspaces in $𝑉={𝔽}_{𝑞}^2$. Since $𝐺$ acts on $𝑉$ by matrix multiplication, it acts on $\mathrm{\Pi }$.

(i) Show that if $\ell \in \mathrm{\Pi }$ then the stabilizer of $\ell$ in $𝐺$ contains a unique $𝑝$-Sylow subgroup of $𝐺$. How many $𝑝$-Sylow subgroups does $𝐺$ have, and what is their order?

(ii) Prove that if ${\ell }_1,{\ell }_2$ and ${\ell }_3$ are three distinct one-dimensional subspaces of $𝑉$, then there is an element $𝑔$ of $𝐺$ such that $𝑔{\ell }_1={𝔽}_{𝑞}\left(\begin{array}{c}0\\ 1\end{array}\right)$, $𝑔{\ell }_2={𝔽}_{𝑞}\left(\begin{array}{c}1\\ 1\end{array}\right)$, $𝑔{\ell }_3={𝔽}_{𝑞}\left(\begin{array}{c}0\\ 1\end{array}\right)$.

(c) Show that if ${𝑃}_1,{𝑃}_2$ and ${𝑃}_3$ are three distinct $𝑝$-Sylow subgroups of $𝐺$, and if ${𝑄}_1,{𝑄}_2$ and ${𝑄}_3$ are another three distinct $𝑝$-Sylow subgroups of $𝐺$, then there exists a $𝑔\in 𝐺$ such that

Let $𝑀$ be a finitely generated nonzero module over a commutative ring $𝑅$ and let $𝐼$ be an ideal of $𝑅$.

(i) Let $𝜙:𝑀\to 𝑀$ be an $𝑅$-module endomorphism with $𝜙(𝑀)\subset 𝐼𝑀$. Prove there are ${𝑎}_{𝑗}\in {𝐼}^{𝑗}$ and $𝑛\ge 1$ with the property that

as endomorphisms of $𝑀$. (Hint: choose generators for $𝑀$ and use "linear algebra.")

(ii) Suppose that $𝐼$ is contained in every maximal ideal of $𝑅$. Prove Nakayama's lemma: $𝐼𝑀\ne 𝑀$.

Let $𝑝$ be a prime, $𝐺={GL}_3(\mathbb{Z}/{𝑝}^5\mathbb{Z})$.

(i) Show that the natural map $𝐺\to {GL}_3(\mathbb{Z}/𝑝\mathbb{Z})$ is surjective, and compute the order of the kernel.

(ii) Compute the size of $𝐺$, and describe an explicit $𝑝$-Sylow subgroup of $𝐺$.

Let $𝐺={SL}_2({𝔽}_3)$. It is a group of order 24.

(i) Show that $𝐺/\{\pm 𝐼\}$ is isomorphic to the alternating group ${𝐴}_4$. Hint: consider the action of $𝐺$ on one dimensional subspaces of ${𝔽}_3^2$ or the conjugation action of $𝐺$ on its set of 3-Sylow subgroups.

(ii) Show that $𝐺$ has three irreducible complex representations of degree 2. Hint: You may use without proof the fact that $𝐺$ has 7 conjugacy classes.

Prove that an algebraically closed field of characteristic zero does not have an automorphism of odd prime order $𝑝$. (Hint: Show that the $𝑝$th roots of unity would belong to the fixed field, and use norms.)